Theorem elmaprdOLD 8896

Description: Obsolete version of elmaprd 8895 as of 30-Aug-2026. (Contributed by Thierry Arnoux, 13-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elmaprdOLD.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
elmaprdOLD.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
elmaprdOLD.3	⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴))

Assertion

Ref	Expression
elmaprdOLD	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Proof of Theorem elmaprdOLD

Step	Hyp	Ref	Expression
1		elmaprdOLD.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴))
2		elmaprdOLD.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		elmaprdOLD.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4	2, 3	elmapd 8885	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m 𝐴) ↔ 𝐹:𝐴⟶𝐵))
5	1, 4	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  ⟶wf 6551  (class class class)co 7440   ↑_m cmap 8872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874

This theorem is used by: (None)